Evaluate: ∫ sin 2x/(sin^4x+cos^4x)dx

Question

Evaluate: \(\int\cfrac{sin\,2x}{(sin^4x+cos^4x)}dx\)

 ∫ sin 2x/(sin⁴x+cos⁴x)dx

Answer 1

To find:\(\int\cfrac{sin\,2x}{(sin^4x+cos^4x)}dx\)

Formula Used:

1. sec² x = 1 + tan² x

\(2.\int\cfrac{1}{1+x^2}dx=tan^{-1}x+c\)

3. sin 2x = 2 sin x cos x

Rewriting the given equation,

\(\int\cfrac{2sin\,x\,cos\,x}{sin^4x+1}\)

Let y = tan x

dy = sec² x dx

Therefore,

\(\int\cfrac{2y}{y^4+1}dy\)

Let z = y²

dz = 2y dy

\(\Rightarrow\int\cfrac{dz}{1+z^2}\)

⇒ tan^-1 z + C

Since z = y²,

⇒ tan^-1(y²) + C

Since y = tan x

⇒ tan^-1(tan² x) + C

Therefore,